where \(B^{(t)} = [B^{(t,T1)},\ldots ,B^{(t,FLAIR)}] \in \mathbb {R}^{g \times p^{(t)}}\)  and \(B^{(t,m)} = [\mbox{$\beta $}_1^{(t,m)};\ldots ;\mbox{$\beta $}_{p^{(t,m)}}^{(t,m)}] \in \mathbb {R}^{g \times p^{(t,m)}}\)  for all \(m \in M\) . Here, \(\mbox{$\beta $}_j^{(t,m)} = (\beta _{1j}^{(t,m)},\ldots ,\beta _{gj}^{(t,m)})^\top \)  for all \(j=1,\ldots ,p^{(t,m)}\)  and \(m \in M\) , and \(vec(\cdot )\)  is the row-wise vectorization of a matrix. The coefficient \(\beta _{kj}^{(t,m)}\)  corresponds to the association of genomic marker \(k\)  and principal component \(j\)  from the imaging sequence \(m\)  in the layer \(t\) . Also, \(I_n\)  is the identity matrix of dimension \(n\) , and \(\otimes \)  denotes the Kronecker product. Note that \(\Delta ^{(t)} \in \mathbb {R}^{p^{(t)} \times p^{(t)}}\)  is a covariance matrix that captures the covariance between the PCs across imaging sequences, which naturally exists as these sequences are imaging the same region/tumor. In other words, although the PC scores corresponding to a specific imaging sequence are uncorrelated (by construction), cross-sequence correlation could still exist (e.g. between PC scores from T1 and PC scores from T2); this correlation is captured through \(\Delta ^{(t)}\) . We can succinctly rewrite \(vec(XB^{(t)}) = Z\mbox{$\beta $}^{(t)}\) , where \(Z = X \otimes I_{p^{(t)}} \in \mathbb {R}^{np^{(t)} \times gp^{(t)}}\)  [1]} and \(\mbox{$\beta $}^{(t)} = vec(B^{(t)}) \in \mathbb {R}^{gp^{(t)}}\) . Let \(\mbox{{$\mathbf {y}$}}^{(t)} = vec(Y^{(t)})\)  and \(\Theta ^{(t)} = I_n \otimes \Delta ^{(t)}\) ; then, Equation (REF ) is equivalent to \(\mbox{{$\mathbf {y}$}}^{(t)} \sim N_{np^{(t)}}(Z\mbox{$\beta $}^{(t)}, \Theta ^{(t)})\) .
